I'm sure a more technical answer will be along soon, but here goes:
The en-route squawks are allocated to NATS en-route by NAS (the national airspace computer). This draws the squawks from either its own database or, if relevant, a number or ORCAM squawks which are allocated to it and which go with the flight until it lands or is allocated a new squawk in another country down the line.
To prevent two aircraft potentially having the same squawk in the same airspace, squawks have a 'life' on them based upon the flight plan. They cannot be reallocated until this life has naturally expired or they are manually marked as available for use (something we have to do if NAS is off-line).
For these reasons, the control of the allocation of en-route squawks has to rest with one body, in this case NATS. In addition, the only real way an airport can get access to these codes 'on-line' is through the provision of a link into NAS. Giving this to non-NATS airports carries a risk of destabilising NAS (it can be a bit vulnerable to data which is not put in in exactly the right format and you don't know it's wrong until NAS falls over and everyone gets asked to stay on the ground!) by personnel who are not trained by or accountable to NATS. It's like giving your internet password to a neighbour and asking them to take care with your logins!!
NATS airports are different because NATS has some influence and control over the training, accountability and level of access of these airports.
Hope this helps a bit.